Definition 2

An A-module M is called pseudo-zero if it is finitely generated and it satisfies the equivalent conditions of Proposition 9.

This definition and Proposition 9 show that a pseudo-zero A-module is a torsion A-module; the converse is false.

Examples (1) If A is a Dedekind domain, every prime ideal of A is of height $\leq 1$; to say that M is pseudo-zero means then that $\operatorname{Supp}(M) = \varnothing$ and hence that $M = 0$ (Chapter 11, $§ 4$, no. 4).

(2) Let $k$ be field and $A = k[X, Y]$ the polynomial ring over $k$ in two indeterminates; if $m$ is the maximal ideal $AX + AY$ of $A$, the $A$-module $A/m$ is pseudo-zero; for its annihilator $m$ is not of height $\leq 1$ since it contains the principal prime ideals $AX$ and $AY$ and is distinct from them; therefore $A : m = A$ (§ 1, no. 6, Corollary 1 to Theorem 3).